The mathematics of dependent types, category theory and entity modelling

The mathematical notions of
(a) Contextual Categories and (b) Generalised Algebraic Theories encapsulate the idea
of
types dependent on contexts which in turn are constructed from types1,2 .
Now types of things are equally concepts and (c) Entity Modelling
as described here is about modelling the types of things of interest in a particular
situation or or as part of a particular endeavour (the chosen perspective); in an
entity model,
composition relationships model dependencies between types of things. (a), (b) and
(c) are therefore closely related.
A project of mine has been to try narrow the gap between (a) and (c) by
finding a variation on the algebra of
contextual categories which, unlike contextual categories, explicitly includes
conjunctive dependencies such as we find them in entity modelling (here on this website
at least) and also in the schemas of network databases.
A paper I published on this in 19863 called
"Formalising the Network and Hierarchical Data Models "
I later came to realise was a long way off the mark.
Subsequently I have come up with revised definitions and these I have drafted here:
DependencyCategories.pdf.

In addition here are some relatively recent notes on
the generalised algebraic theory of contextual categories:
theGATofCCs.pdf. I wrote these after having learnt of the definition of C-system given
by Vladimir Voevodsky.

Finally, here is a recent write up of some work that I did at the time of my thesis,
basically it is a variation
on contextual categories — in brief, meta-GAT algebras are to a contexual category
as a clones are to Lawvere theories.
I wrote these notes suspecting, as was subsequently confirmed, that Voevodsky was
producing a paper describing similar structures to my meta-GAT algebras (he calls
them B-systems):
MetaGAT and MetaGAT algebras.pdf.